if z1 and z2 are two non-zero complex numbers such that|z1|=|z2|+|z1-z2| then prove that Im(z1/z2) =0

To tackle the problem you've presented, we start by unpacking the equation given: |z1| = |z2| + |z1 - z2|. This equation involves the magnitudes of two complex numbers, z1 and z2, and we want to show that the imaginary part of the quotient z1/z2 is zero, meaning that z1 and z2 are aligned along the same direction in the complex plane. Let’s break this down step by step.

Understanding the Magnitudes

The expression |z1| represents the distance from the origin to the point in the complex plane represented by z1, and similarly for |z2|. The term |z1 - z2| represents the distance between the points z1 and z2. The equation states that the distance from the origin to z1 is equal to the distance from the origin to z2 plus the distance between z1 and z2.

Geometric Interpretation

Geometrically, this can be interpreted as follows: if you visualize the points corresponding to z1 and z2 in the complex plane, the equation indicates that the point z1 lies on the circle centered at the origin with radius |z1|, while z2 is positioned such that the distance from the origin to z2 plus the distance from z2 to z1 equals the distance from the origin to z1. This can only happen if the points z1, z2, and the origin are collinear.

• If z2 is on the line segment connecting the origin to z1, then the angles made with the real axis are the same for both z1 and z2.
• This means that z1 and z2 must be scalar multiples of each other, which can be expressed as z1 = k * z2 for some positive real number k.

Expressing z1 and z2

Let’s denote z1 as k * z2, where k is a positive real number. Now we can write:

z1/z2 = (k * z2) / z2 = k.

Since k is a positive real number, we can see that:

Im(z1/z2) = Im(k) = 0.

Conclusion

Thus, we have shown that if |z1| = |z2| + |z1 - z2|, then the imaginary part of z1/z2 is indeed zero. This result confirms that z1 and z2 must lie on the same line through the origin, reinforcing our understanding of the relationship between their magnitudes and their positions in the complex plane. Therefore, the imaginary part of the quotient of these two complex numbers is zero, indicating that they are collinear.